# Are decimals flawed

#### cheesy999

First: a maths problem i'm sure most of us know - 0.99 recurring x 10 - 0.99 recurring =9

this states that 0.99 recurring is = 1 and so does the equation

1/3=0.3'

however when you multiply out 0.3' you get 0.9' which means that 0.9' and 1 are the same, however 0.9' is not 1, but slightly less

my understanding of maths however, is that one numerical value cannot be the same as another

does this mean maths is flawed?

if so, is there a number system under which these problems do not occur?

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#### lucas4

##### New Member
no, because recurring means there is no end to the number (IIRC)
u can't do it on a calculator or anything, because when ur typing in the number, it isnt actually recurring unless u type it as a fraction.
i see what you are saying, but the margin of error is so minute, its practically insignificant.

at least thats the way i see it !

interesting point none the less

#### lucas4

##### New Member
First: a maths problem i'm sure most of us know - 9.99 recurring x 10 - 9.99 recurring =9

this states that 0.99 recurring is = 1 and so does the equation
it = 90 btw

#### lucas4

##### New Member
its an interesting point, and quite awkward to explain why 0.99 doesnt = 1.

i think the easiest way to explain it may be rounding error in the apparatus we use ??

#### Fourstaff

##### Moderator
Staff member
Studying for a maths major here, so I think I have some credibility.

We know that 0.9 cannot be bigger than 1. Fix 0.9 recurring as a sequence, calling it a(n), where a(1) = 0.9, a(2) = 0.99 etc. Now we make this statement: For any positive number x, there exist an N such that N is a natural number (positive integer) and when n>N |1-a(N)|< x . From that statement, its obvious that a(n) tends to 1, therefore 0.99 recurring is 1.

Analysis is a bitchy subject, and unless your job/life/education depends on it, its best avoided. We do silly things like finding the limit of 0/0 (0/0 tends to some random things depending on how you define the top and bottom zero).

#### cheesy999

its an interesting point, and quite awkward to explain why 0.99 doesnt = 1.

i think the easiest way to explain it may be rounding error in the apparatus we use ??
not a rounding error

if x = 0.9'
therefore 10x = 9.9'
10x-x=9x = 9.9' - x

therefore 9x = 9

#### mlee49

It's all about being accurate as you need to be.

Decimals approximations are great for whole number divisions but when you try to approximate pi with decimals you'll fall short every time. However if you'd like to be accurate simply choose your degree of accuracy and there's a fraction to go with it! 22/7 or 355/113 ... it goes on!

#### Fourstaff

##### Moderator
Staff member
not a rounding error

if x = 0.9'
therefore 10x = 9.9'
10x-x=9x = 9.9' - x

therefore 9x = 9
10x - x = 9x =/= 9.9' - x
9.9' - x = 9.9' - 0.9' = 9

#### cheesy999

It's all about being accurate as you need to be.

Decimals approximations are great for whole number divisions but when you try to approximate pi with decimals you'll fall short every time. However if you'd like to be accurate simply choose your degree of accuracy and there's a fraction to go with it! 22/7 or 355/113 ... it goes on!
my point is

why use decimals if their not good?

i use fractions myself but because there a lot easier

#### Fourstaff

##### Moderator
Staff member
my point is

why use decimals if their not good?

i use fractions myself but because there a lot easier
Decimal is good, but it can only do so much. Fractions are neat and easy to work with, but they are much less accurate than decimals.

my sig has all the answers you need
Your sig proves the => but not the <= which is what we want. You might as well say W1zzard is either a guy or a girl or both.

#### cheesy999

Decimal is good, but it can only do so much. Fractions are neat and easy to work with, but they are much less accurate than decimals.
how are they accurate if the system has errors?

i mean - 1/3 is precisely a 1/3 of the object

#### Fourstaff

##### Moderator
Staff member
how are they accurate if the system has errors?

i mean - 1/3 is precisely a 1/3 of the object
The system has no errors

#### Frick

##### Fishfaced Nincompoop
how are they accurate if the system has errors?

i mean - 1/3 is precisely a 1/3 of the object
That is not an error. It's an infinite number so you can't express it in decimals.

I don't really get your point btw. Since when is 0.99 recuring = 1? And 1/3 has never been 0.3. At least not in my schools.

#### cheesy999

That is not an error. It's an infinite number so you can't express it in decimals.
sounds like a problem with the system if you can't use it for something

i mean you couldn't do one calculation on an pentium processor and that was counted as an error - http://en.wikipedia.org/wiki/Pentium_FDIV_bug

there's one problem with a calculation in decimals - doesn't that mean their broken?

#### Fourstaff

##### Moderator
Staff member
sounds like a problem with the system if you can't use it for something

i mean you couldn't do one calculation on an pentium processor and that was counted as an error - http://en.wikipedia.org/wiki/Pentium_FDIV_bug

there's one problem with a calculation in decimals - doesn't that mean their broken?
You will need to work with symbolic to ensure no rounding error problems, and x86 does not do its math symbolically. Well, in that sense fractions is better than decimals, contradicting what I said earlier :/

#### Kreij

##### Senior Monkey Moderator
1/3 is not a number, it is a divisional proportion.
All fractions are estimates based on what they are derived from, and when converted to their decimal equivelents are rounded as to be useful.

When you do something 1/2 assed can it be equated to 0.5 assed? Not necessailry.

#### lucas4

##### New Member
sounds like a problem with the system if you can't use it for something

i mean you couldn't do one calculation on an pentium processor and that was counted as an error - http://en.wikipedia.org/wiki/Pentium_FDIV_bug

there's one problem with a calculation in decimals - doesn't that mean their broken?
i dont think it means they broken.

they are fit for their purpose, and we all know that they cannot be 100% accurate when we have a recurring answer or pi.
this is why we round our answers to a certain number of decimal places which is considered to be an accurate enough margin.

eg, sometimes rounding earlier answers to 4dp if u need a 2dp final answer is perfectly acceptable level of accuracy, whereas rounding to 4dp if we need a 10dp final answer isnt.
we know when to use the appropriate degree of accuracy which doesnt make them broken IMO

#### Fourstaff

##### Moderator
Staff member
1/3 is not a number, it is a divisional proportion.
All fractions are estimates based on what they are derived from, and when converted to their decimal equivelents are rounded as to be useful.
An engineer's point of view. We can sit here all day arguing who is right, but I have weapons in my arsenal to make me right.

#### lucas4

##### New Member
An engineer's point of view. We can sit here all day arguing who is right, but I have weapons in my arsenal to make me right.
no need for a gun fight

#### cheesy999

1/3 is not a number, it is a divisional proportion.
All fractions are estimates based on what they are derived from, and when converted to their decimal equivelents are rounded as to be useful.

When you do something 1/2 assed can it be equated to 0.5 assed? Not necessailry.
so is there such thing as a number system where things like this are not a problem (say Roman numerals/attic numerals for example)

#### lucas4

##### New Member
so is there such thing as a number system where things like this are not a problem (say Roman numerals/attic numerals for example)
*swoosh* now the talk goes above my head

#### Frick

##### Fishfaced Nincompoop
IMO numbers and math is above such worldly things as usefullness. Just because we can't pin them down doesn't mean they're not useful. Think about pi.